Sooo I'm not seeing it described how I would describe it.
22 times 40
becomes
22
x40
You start by multiplying the digit in the ones place of the lower number by the ones place of the upper number. The result is written below.
22
x40
0
Now you multiply the bottom ones place by the next digit of the upper number, which is the tens place, and write that result below.
22
x40
00
The first answer row is complete because we have multiplied the bottom number ones digit by the whole upper number.
Now we need to start a new answer row, below the first one, and every time you add a new answer row you lead it with a fresh 0 to shift it to the left. Here is our problem with the new zero.
22
x40
00
0
We are now engaging the "4" which is the next number to the left of the bottom row of the problem, the tens place. We need to multiply that "4" by the "22" above. Again we do it in two steps, starting with multiplying the "4" with the rightmost "2" in the upper number's ones place.
22
x40
00
80
Then we multiply the "4" with the "2" in the tens place on the top row.
22
x40
00
880
In all cases we're loading the new digit to the left in our answer row until we run out of digits in the top row to multiply.
Right now we would check to see if there are any more digits in the "40" row to continue the process. There aren't. We're finished multiplying.
Now we add everything in our answer row, vertically.
22
x40
00
+880
0
22
x40
00
+880
80
22
x40
00
+880
880
And there is our answer.
There's an extra step if we're multiplying with digits large enough to make an individual multiplication give an answer over 9.
Let's say we start with 23 x 85.
23
x85
The first step would be multiplying the 1s digit of the lower number by the 1s digit of the upper number.
23
x85
15
But then the next space we should enter an answer digit is already taken up by that 1! Such an unscrupulous 1, sitting in someone else's seat. Instead of writing it in the answer row, we carry it up to above the next top-row digit - the tens place.
1
23
x85
5
Now we do our next multiplication ...
1
23
x85
105
... and then add the carried 1 from above to the answer ...
1
23
x85
115
... and then dang, we're over one digit, so we have to carry again like so.
11
23
x85
15
This next part requires that you make a leap - of imagination! Any number actually has zeroes in front of it. So if I write "25" it's actually equal to "000025". This is because I'm really saying "this is a number with 5 in the ones place, 2 in the tens place, 0 in the hundreds place, 0 in the thousands place" etc. But normally it doesn't matter that there's nothing in all those higher digit places - nobody cares because they don't do any work. WELL THEY DO NOW BUDDY.
You're going to imagine there's a zero in front of our lovely "23" up there. You multiply like so (except without actually writing out the zero in real life, because we're too cool for that):
11
023
x85
015
Here we just said "5 times 0 equals zero", but we need to add the carried "1" to that result, which makes it just "1" below.
11
023
x85
115
Great! Excellent! Done with the first part. We've multiplied the 1s place of the lower number by the entire upper number. You may ask, "How did you know to stop adding zeroes and keep multiplying forever? I stopped because I knew any more zeroes would just result in an answer of 0 again, because 5 x 0 = 0 and there are no more carried leftovers above to add up to anything.
Now let's start on the lower row. LEAD WITH A ZERO because we're starting a new answer row. Also I'm crossing out all my carries up top so I don't get confused, and writing all my new carried numbers above it. In handwriting you'd use tiny numbers but I tried and the forum makes them look confusing.
Here's what it looks like after I struck out my old carried digits, which we won't use again, and add that leading zero to my second answer row.
11
23
x85
115
0
Now I multiply the top row using the tens place digit of the bottom row, starting with the top row's 1s place.
2
11
23
x85
115
40
You'll notice for convenience I stick the carried digit above the next thing I'll be multiplying.
12
11
23
x85
115
840
Here I said "8 x 2 = 16, and add the carried 2, =18. Of that 18, the 8 goes in the answer row and I carry the 1 to the row above."
Now we again have a carried digit and nothing to multiply. I'll add that "phantom zero" to show what's happening: 8 x 0 = 0, add that carried 1, the answer is 1, which we add to the left side of the answer row.
12
11
023
x85
115
1840
Now that we're out of multiplication to do, we add.
12
11
23
x85
115
+1840
5
And Then
12
11
23
x85
115
+1840
55
But Then
12
11
23
x85
115
+1840
955
And So (RETURN OF THE PHANTOM ZERO)
12
11
23
x85
0115
+1840
1955
In short, there is a Rule in multiplication that says you can break up and mix up stuff and still multiply it as long as you do it exactly right. Long multiplication like this is a way of making sure you get it right. Another way to explain it is that when you multiply 11 x 5, what you're actually doing, is multiplying 10 x 5 and also 1 x 5 and then adding the result together. If you wanted to multiply 111 x 5, you would do 100 x 5 plus 10 x 5 plus 1 x 5. As long as you keep the digits you're multiplying in exactly the right places (as in, ones place, tens place, etc.) it will work out. And the reason you would do it that way? Because we know that multiplying by zero equals zero! So I can easily skip past a bunch of steps and spot 500 + 50 + 5 right away.
When you want to multiply a two digit number by a two digit number, you're splitting it up into a whole lot of individual multiplications. You're taking, say, 11 x 11, and doing the following:
10 x 1 = 10
1 x 1 = 1
10 x 10 = 100
1 x 10 = 10
And adding the results together (121, which is what you get with a calculator or long multiplication doing 11 x 11). Again, long multiplication is a way to make sure you didn't miss any of those tasks - such as when multiplying a ten digit number by another ten digit number. It also saves space, believe it or not.
Don't be surprised if stupid math tricks don't work for you until you go through algebra. It changes the way you think about some things and the processes you learn can really help your basic math.
Let's say you have two numbers you want to multiply, like 55 x 12. I can break that down in my head into two operations that are simpler: 55 x 10 and 55 x 2.
The 55 x 10 is super easy because it follows a kinda pseudo-rule that I don't think they bother to put into the textbooks until algebra: multiplying by 10, just add a zero to the right side of the other number. If you need to multiply by some larger one like 10,000 just strip the zeroes off one and stick em on the other: 55 x 10,000 means strip four zeroes off the right side and stick them on the left. So you end up with 550,000.
Anyway, that means our first part is 550. What about 55 x 2? I need to break that down into 50 x 2 and 5 x 2. Suddenly it's clear: 50 x 2 = 100, and 5 x 2 = 10, so you add them together and get 110. Now we go back up and add the two answer results: 550 + 110 = 660.
With certain numbers you can do this in your head pretty well. It's kinda tough to do unless you have a good handle on your higher-end multiplication table. Also it's easier for me to do when I imagine a sheet of paper with the numbers written on it - although your method may vary.
The boxes method is great but only if you have a handle on that multiplication trick I mentioned above with the 10s. This is because you'll need to be able to figure out in your head what 60 x 50 is for example. This requires an extra process: stripping away the two 0s, multiplying 6 x 5, and then slapping the two zeroes back on at the end. Like so:
60 x 50
6 x 5 (with 00 off to the side)
30 (with 00 off to the side)
3000
Which kinda requires knowing the various Rules of Multiplication that they don't teach until algebra.
I suspect Boxes may be fast if you're good at it, but so is long multiplication, and both get you to the same place.
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Topic: multiplication easy...WHAT?!? (Read 3912 times)